ax^{2 }+^{ }bx + c = 0

By

James
W. Wilson and Audrey V. Simmons

University of Georgia

It has now become a rather standard exercise, with available technology, to construct graphs to consider

the equation ax^{2} + bx + c = 0. We will be looking at some of the
relationships that exit in the world of parabolas.

Our starting point becomes y
= ax^{2} + bx + c

For **a = 1**, **c = 1** our equation is y = x^{2} + bx + c. We will vary **b**.

Let -3 **b** 3

y= x^{2}
+1 y= x^{2} + 1x +
1 y=
x^{2} +2x +1 y= x^{2}
+3x +1 y= x^{2} -1x +1

y= x^{2} –2x + 1 (Yellow) y= x^{2} –3x + 1 (gray)

We can see that all of the
graphs have a y-intercept of (0,1).
Would we have a y- intercept of 3 if **c=3** and the equation was y=x^{2 }+ bx +3 ?

Click ** HERE** to
see if there is a common y-intercept.

Would the same be true when **a=2** and **c=3**
for y= 2x^{2} +bx + 3?

Click HERE
to see.

Surprise! No matter what the coefficients of a or
b, the value of **c **tells
us what the y-intercept is. There
is a movement of the parabolas through the same point on the y-axis.

Please look back at the
original graphs above. At the
points where b=2 and b= - 2 the graphs are tangent to the x- axis. Recall that
the roots of the equation indicate where the graph touches or crosses the x
-axis.

The roots of the equations
are as follows:

y
= x^{2} +1
no roots

y
= x^{2} + x + 1
no roots y
= x^{2} -1x + 1 no roots

y
= x^{2} + 2x + 1 one
root y
= x^{2} -2 x + 1 one
root

y
= x^{2} + 3x + 1 two
roots y
= x^{2} -3 x + 1 two
roots

When b> 2, there are two
negative real roots. When b<-2,
there are two positive real roots.
When –2< b < 2, there are no real roots.

Is the locus of the vertices
of a set of parabolas a parabola?

Look at the black graph
of y = -x^{2} +1. It appears to travel through the
vertices of each parabola.

If we found the vertices of
each of the graphs, they would be solutions for the equation y = -x^{2} +1

Consider the same equation x^{2}
+bx +1 = 0. We will graph this
equation in the **xb** plane. That means we will solve for b instead
of y giving us the equation

b = or b =

**x** will still be found on the horizontal axis and **b** will be found on the “y” axis.

What are the roots of the
graph? Our quadratic equation x^{2} +bx +1 = 0 graphs as a hyperbola.
When b = 2 or b =
-2 we have one root at the vertex of the hyperbola. When b = 3
we have 2 negative roots. Therefore, for all values of b > 2, there are 2
negative roots. When b < -2, there will be 2 positive roots.

When c =-1, values less than
–1 approach a diagonal asymptote and the vertical axis.

This time we will graph our
equation in the **xc** plane. Our equation is

x^{2} + 5x +c =
0 or c = -x^{2}
– 5x

This will be the graph of a
parabola. In the graph below **x** is represented on the horizontal axis and **c** is represented on the vertical axis.

x^{2} +
5x +c = 0 is our parabola

c = 6.25 shows
one root

c> 6.25 shows no roots

c=1 shows two negative roots

c
= 0 shows one negative root and one root of zero

c = -2 shows one positive and one negative root

To summarize when c < 0 there are two roots, one is positive and one is negative. When c = 0, one root is negative and one is zero. When 0< c < 6.25, there are two negative roots. There is only one root when c = 6.25.